Elena Cherkaeva

Inverse electromagnetic scattering models for sea ice

Inverse scattering algorithms for reconstructing the physical properties of sea ice from scattered electromagnetic field data are presented. The development of these algorithms has advanced the theory of remote sensing, particularly in the microwave region, and has the potential to form the basis for a new generation of techniques for recovering sea ice properties, such as ice thickness, a parameter of geophysical and climatological importance. Moreover, the analysis underlying the algorithms has led to significant advances in the mathematical theory of inverse problems. In particular, the principal results include the following. (1) Inverse algorithms for reconstructing the complex permittivity in the Helmholtz equation in one and higher dimensions, based on layer stripping and nonlinear optimization, have been obtained and successfully applied to a (lossless) laboratory system. In one dimension, causality has been imposed to obtain stability of the solution and layer thicknesses can be obtained from the recovered dielectric profile, or directly from the reflection data through a nonlinear generalization of the Paley-Wiener theorem in Fourier analysis. (2) When the wavelength is much larger than the microstructural scale, the above algorithms reconstruct a profile of the effective complex permittivity of the sea ice, a composite of pure ice with random brine and air inclusions. A theory of inverse homogenization has been developed, which in this quasistatic regime, further inverts the reconstructed permittivities for microstructural information beyond the resolution of the wave. Rigorous bounds on brine volume and inclusion separation for a given value of the effective complex permittivity have been obtained as well as an accurate algorithm for reconstructing the brine volume from a set of values. (3) Inverse algorithms designed to recover sea ice thickness have been developed. A coupled radiative transfer-thermodynamic sea ice inverse model has accurately reconstructed the growth of a thin, artificial sea ice sheet from time-series electromagnetic scattering data.

Inverse bounds for microstructural parameters of composite media derived from complex permittivity measurements

Abstract Bounds on the volume fraction of the constituents in a two-component mixture are derived from measurements of the effective complex permittivity of the mixture, using the analyticity of the effective property. First-order inverse bounds for general anisotropic materials, as well as second-order bounds for isotropic mixtures, are obtained. By exploiting an analytic representation of the effective complex permittivity, the problem of estimating the structural parameters is reduced to a problem of evaluating the moments and support of a measure containing information about the geometrical structure of the material. Rigorous bounds on the volume fraction are found by inverting first- and second-order (Hashin–Shtrikman) forward bounds on the complex permittivity. The inverse bounds are applied to measurements of the effective complex permittivity of sea ice, which is a three-component mixture of ice, brine and air. The sea ice is treated via the two-component theory applied to a mixture of brine and a...

Optimal survey design using focused resistivity arrays

The first problem which needs to be solved when planning any geoelectrical survey is a choice of a particular electrode configuration that can give the maximal response from a target inhomogeneity. The authors formulate a problem of maximizing the response as an optimization problem for an applied current intensity distribution on the surface. The solution of this problem is the optimal intensity distribution of the current, which maximizes the response from the inclusion. This problem is solved numerically with singular value decomposition of an impedance matrix. The optimal current array is modeled as a current of varying optimal intensity injected at different electrodes. The problem does not need any information about the inclusion but its measured impedance matrix. Thus an optimal current array can be designed for every particular resistivity distribution. The optimal current patterns are found for a number of models of a conductive inclusion, and responses due to the optimal current are compared with responses due to conventional arrays. This method can be applied to any background and inclusion resistivity distribution.

Inverse conductivity problem for inaccurate measurements

In order to determine the conductivity of a body or of a region of the Earth using electrical prospecting, currents are injected on the surface, surface voltage responses are measured, and the data are inverted to a conductivity distribution. In the present paper a new approach to the inverse problem is considered for measurements containing noise in which data are optimally chosen using available a priori information at the time of the imaging. For inexact data the eigenvalues of the current-to-voltage boundary mapping show what part of the conductivity function can be confidently restored from the measurements. A special choice of measurements permits simple inversion algorithms reconstructing only this reliable part of the solution, hence reducing the dimension of the inverse problem. The inversion approach is illustrated in application to numerical modelling via a very fast approximate imaging solution.

Structural Optimization and Biological “Designs”

The amazing rationality of biological “constructions” excites the interest to modelling them by using the mathematical tools developed in the theory of structural optimization. The structural optimization solves a geometrical problem of the “best” displacements of different materials in a given domain, under certain loadings. Of course, this approach simplifies the real biological problem, because the questions of the mechanism of the building and maintaining of structures are not addressed. The main problem is to guess a functional for the optimization of a living organism. The optimal designs are highly inhomogeneous; their microstructures may be geometrically different, but possess the same effective properties. Therefore the comparing of the various optimal geometries is not trivial. We show, that the variety of optimal geometries shares the same characteristics of the stress tensor in any optimal structure. Namely, special norm of this tensor stay constant within each phase of the optimal mixture.

High-resolution adaptive induction logging: petrophysical and electromagnetic considerations

Incorporation of independent formation information into inductive log interpretation will become more important as the resolution demands on induction logging increase. Often, such information can consist of a reasoned petrophysical characterization of the target and its inductive signature. This target characterization, together with other information, can be used to adaptively focus array tools as a function of borehole depth.